10.4 Moments of Inertia About Inclined Axes; Principal Moments

10.4 Moments of Inertia About Inclined Axes; Principal Moments

10.4 Moments of Inertia About Inclined Axes; Principal Moments Example 1, page 1 of 3

1. Determine the moments of inertia of the standard rolled-steel angle section with respect to the u and v axes.

y

0.987 in.

0.5 in.

Ix = 17.40 in4

v

Iy = 6.27 in4

Ixy = 6.08 in.4

6 in.

C 45? u

4 in.

x 1.99 in. 0.5 in.

10.4 Moments of Inertia About Inclined Axes; Principal Moments Example 1, page 2 of 3

1 The formula for Iu is

Iu = Ix + Iy + Ix Iy cos 2

2

2

Ixy sin 2

(1)

We can save some work later, if we calculate and save the expressions

Ix + Iy 2

=

17.40 in4 + 6.27 in4 2

= 11.835 in4

(2)

and

Ix Iy 2

=

17.40 in4 2

6.27 in4

= 5.565 in4

(3)

Eq. 1 becomes,

11.835 in4, by Eq. 2 6.08 in.4

45? ( is negative because the x axis

Iu =

Ix + Iy 2

+ Ix Iy cos 2 2

Ixy sin 2

must be rotated clockwise to make it coincide with the u axis)

5.565 in4, by Eq. 3

= 5.76 in4

Ans.

10.4 Moments of Inertia About Inclined Axes; Principal Moments Example 1, page 3 of 3

2 Similarly, for Iv and Iuv,

11.835 in4, by Eq. 2

6.08 in.4

45?

Iv =

Ix + Iy 2

Ix Iy cos 2 2

Ixy sin 2

5.565 in4, by Eq. 3

= 17.92 in4

Ans.

5.565 in4, by Eq. 3

45?

Iuv = Ix 2 Iy sin 2

Ixy cos 2

= 5.57 in4

6.08 in.4 Ans.

10.4 Moments of Inertia About Inclined Axes; Principal Moments Example 2, page 1 of 5

2. Determine the moments of inertia of the crosshatched area with respect to the u and v axes for a) = 25? and b) = 90?

40 mm

y v

220 mm u

x

100 mm

100 mm

20 mm

1 Before we can use the equations for Iu, Iv, and Iuv, we must determine Ix, Iy, and Ixy. Determining Ixy is easy: the y axis is an axis of symmetry, so

Ixy = 0

(1)

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